Abstract
The soft-pion theory and the Fubini-Furlan mass dispersion relations have been used to analyze the pion-nucleon scattering lengths and obtain a value for the $\ensuremath{\sigma}$ commutator term. With this value and using the same principles, scattering lengths have been predicted for nuclei with mass number ranging from 6 to 23. Agreement with experiment is very good. For those who believe in the Gell-Mann-Levy $\ensuremath{\sigma}$ model, the evaluation of the commutator yields the value $0.26{(\frac{{m}_{\ensuremath{\sigma}}}{{m}_{\ensuremath{\pi}}})}^{2}$ for the $\ensuremath{\sigma}$-nucleon coupling constant. The large dispersive corrections for the isosymmetric case imply that the basic idea behind many of the soft-pion calculations, namely, slow variation of matrix elements from the soft-pion limit to the physical pion mass, is not correct.
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